Can negative absolute temperature be achieved?

Your question seems to be predicated on the idea that negative absolute temperature is somehow supposed to be "colder" than absolute zero, and you are right that that would be non-sensical.

But actually, in a very precise sense, negative absolute temperatures are hotter than all positive temperatures, see also this question. This is simply a result of the statistical definition of temperature $T$, which is $$ \frac{1}{T} = \frac{\partial S}{\partial E},$$ where $S$ is the entropy of the system and $E$ its energy content, so $\beta := \frac{1}{T}$ is actually the more natural quantity to think about in the physical formalism. See also this excellent answer by DanielSank for how the $\frac{1}{T}$ appears naturally as a Lagrange multiplier in thermodynamics.

From the above, we see that temperature is negative for system whose entropy decreases as the energy rises. Such systems are unusual, but they are not forbidden. Absolute zero, $T\to 0$, corresponds to $\beta \to \infty$. As the system gets hotter, $\beta$ decreases. $\beta\to 0$ looks weird in terms of temperature, since it corresponds to $T\to \infty$, but since it is $\beta$ that is of primary physical importance, it is not actually forbidden for it to cross zero and become negative. In terms of temperature, a system crossing the $\beta = 0$ point would have to be described as heating up all the way to "infinite positive temperature", then flipping the sign of the temperature and starting to go towards $T= 0$ again from $-\infty$.

To expand on ACuriousMind's post, consider as a toy model a system with $2$ energy levels, say $0$ and $E>0$. These have respective probabilities $(1+e^{-\beta E})^{-1},\,e^{-\beta E}(1+e^{-\beta E})^{-1}$, so the mean energy is $E(1+e^{\beta E})^{-1}$. Clearly, this is $0$ at $\beta=+\infty$ ($T=0^+$), $E/2$ at $\beta=0$ ($T=\infty$) and $E$ at $\beta=-\infty$ ($T=0^-$), so a large negative $T$ is especially hot as measured by mean energy.

It turns out $2$-level systems cannot reach $\beta < 0$ while in thermal equilibrium. Lasers work by maintaining negative $T$ in systems with at least $3$ levels (well, most do). Thus the highest energy level is the most occupied. This is an example of population inversion.